Survival, Absorption, and Escape of Interacting Diffusing Particles

We study the probability distribution P(A) of the area A =
R T
0
x(t)dt swept under fractional
Brownian motion (fBm) x(t) until its first passage time T to the origin. The process starts at t = 0
from a specified point x = L. We show that P(A) obeys exact scaling relation
P(A) = D
1
2H
L
1+ 1
H
ΦH

D
1
2H A
L
1+ 1
H
!
,
where 0 < H < 1 is the Hurst exponent characterizing the fBm, D is the coefficient of fractional
diffusion, and ΦH(z) is a scaling function. The small-A tail of P(A) has been recently predicted
by Meerson and Oshanin [Phys. Rev. E 105, 064137 (2022)], who showed that it has an essential
singularity at A = 0, the character of which depends on H. Here we determine the large-A tail of
P(A). It is a fat tail, in particular such that the average value of the first-passage area A diverges
for all H. We also verify the predictions for both tails by performing simple-sampling as well as
large-deviation Monte Carlo simulations. The verification includes measurements of P(A) up to
probability densities as small as 10−190. We also perform direct observations of paths conditioned
to the area A. For the steep small-A tail of P(A) the “optimal paths”, i.e. the most probable
trajectories of the fBm, dominate the statistics. Finally, we discuss extensions of theory to a more
general first-passage functional of the fBm.